# Properties

 Label 2304.4.a.t.1.1 Level $2304$ Weight $4$ Character 2304.1 Self dual yes Analytic conductor $135.940$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,4,Mod(1,2304)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2304.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2304.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$135.940400653$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 384) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 2304.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-18.4222 q^{5} +22.4222 q^{7} +O(q^{10})$$ $$q-18.4222 q^{5} +22.4222 q^{7} -53.6888 q^{11} -7.15559 q^{13} -39.6888 q^{17} +125.689 q^{19} -99.1556 q^{23} +214.378 q^{25} +205.800 q^{29} +147.489 q^{31} -413.066 q^{35} -125.689 q^{37} +506.444 q^{41} -413.689 q^{43} +313.911 q^{47} +159.755 q^{49} -44.3331 q^{53} +989.066 q^{55} +324.000 q^{59} -324.000 q^{61} +131.822 q^{65} -464.266 q^{67} -1052.84 q^{71} +1022.27 q^{73} -1203.82 q^{77} -602.910 q^{79} +15.8217 q^{83} +731.156 q^{85} -381.378 q^{89} -160.444 q^{91} -2315.47 q^{95} +659.154 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{5} + 16 q^{7}+O(q^{10})$$ 2 * q - 8 * q^5 + 16 * q^7 $$2 q - 8 q^{5} + 16 q^{7} + 8 q^{11} - 72 q^{13} + 36 q^{17} + 136 q^{19} - 256 q^{23} + 198 q^{25} + 152 q^{29} - 80 q^{31} - 480 q^{35} - 136 q^{37} + 436 q^{41} - 712 q^{43} + 224 q^{47} - 142 q^{49} + 344 q^{53} + 1632 q^{55} + 648 q^{59} - 648 q^{61} - 544 q^{65} + 456 q^{67} - 2048 q^{71} + 660 q^{73} - 1600 q^{77} + 496 q^{79} - 776 q^{83} + 1520 q^{85} - 532 q^{89} + 256 q^{91} - 2208 q^{95} - 1220 q^{97}+O(q^{100})$$ 2 * q - 8 * q^5 + 16 * q^7 + 8 * q^11 - 72 * q^13 + 36 * q^17 + 136 * q^19 - 256 * q^23 + 198 * q^25 + 152 * q^29 - 80 * q^31 - 480 * q^35 - 136 * q^37 + 436 * q^41 - 712 * q^43 + 224 * q^47 - 142 * q^49 + 344 * q^53 + 1632 * q^55 + 648 * q^59 - 648 * q^61 - 544 * q^65 + 456 * q^67 - 2048 * q^71 + 660 * q^73 - 1600 * q^77 + 496 * q^79 - 776 * q^83 + 1520 * q^85 - 532 * q^89 + 256 * q^91 - 2208 * q^95 - 1220 * q^97

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −18.4222 −1.64773 −0.823866 0.566785i $$-0.808187\pi$$
−0.823866 + 0.566785i $$0.808187\pi$$
$$6$$ 0 0
$$7$$ 22.4222 1.21069 0.605343 0.795965i $$-0.293036\pi$$
0.605343 + 0.795965i $$0.293036\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −53.6888 −1.47162 −0.735809 0.677190i $$-0.763198\pi$$
−0.735809 + 0.677190i $$0.763198\pi$$
$$12$$ 0 0
$$13$$ −7.15559 −0.152662 −0.0763309 0.997083i $$-0.524321\pi$$
−0.0763309 + 0.997083i $$0.524321\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −39.6888 −0.566233 −0.283116 0.959086i $$-0.591368\pi$$
−0.283116 + 0.959086i $$0.591368\pi$$
$$18$$ 0 0
$$19$$ 125.689 1.51763 0.758816 0.651306i $$-0.225778\pi$$
0.758816 + 0.651306i $$0.225778\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −99.1556 −0.898929 −0.449465 0.893298i $$-0.648385\pi$$
−0.449465 + 0.893298i $$0.648385\pi$$
$$24$$ 0 0
$$25$$ 214.378 1.71502
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 205.800 1.31780 0.658898 0.752232i $$-0.271023\pi$$
0.658898 + 0.752232i $$0.271023\pi$$
$$30$$ 0 0
$$31$$ 147.489 0.854508 0.427254 0.904132i $$-0.359481\pi$$
0.427254 + 0.904132i $$0.359481\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −413.066 −1.99489
$$36$$ 0 0
$$37$$ −125.689 −0.558463 −0.279231 0.960224i $$-0.590080\pi$$
−0.279231 + 0.960224i $$0.590080\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 506.444 1.92910 0.964552 0.263892i $$-0.0850063\pi$$
0.964552 + 0.263892i $$0.0850063\pi$$
$$42$$ 0 0
$$43$$ −413.689 −1.46714 −0.733569 0.679615i $$-0.762147\pi$$
−0.733569 + 0.679615i $$0.762147\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 313.911 0.974226 0.487113 0.873339i $$-0.338050\pi$$
0.487113 + 0.873339i $$0.338050\pi$$
$$48$$ 0 0
$$49$$ 159.755 0.465759
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −44.3331 −0.114898 −0.0574492 0.998348i $$-0.518297\pi$$
−0.0574492 + 0.998348i $$0.518297\pi$$
$$54$$ 0 0
$$55$$ 989.066 2.42483
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 324.000 0.714936 0.357468 0.933925i $$-0.383640\pi$$
0.357468 + 0.933925i $$0.383640\pi$$
$$60$$ 0 0
$$61$$ −324.000 −0.680065 −0.340032 0.940414i $$-0.610438\pi$$
−0.340032 + 0.940414i $$0.610438\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 131.822 0.251546
$$66$$ 0 0
$$67$$ −464.266 −0.846554 −0.423277 0.906000i $$-0.639120\pi$$
−0.423277 + 0.906000i $$0.639120\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −1052.84 −1.75985 −0.879927 0.475109i $$-0.842409\pi$$
−0.879927 + 0.475109i $$0.842409\pi$$
$$72$$ 0 0
$$73$$ 1022.27 1.63900 0.819501 0.573078i $$-0.194251\pi$$
0.819501 + 0.573078i $$0.194251\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −1203.82 −1.78167
$$78$$ 0 0
$$79$$ −602.910 −0.858642 −0.429321 0.903152i $$-0.641247\pi$$
−0.429321 + 0.903152i $$0.641247\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 15.8217 0.0209236 0.0104618 0.999945i $$-0.496670\pi$$
0.0104618 + 0.999945i $$0.496670\pi$$
$$84$$ 0 0
$$85$$ 731.156 0.933000
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −381.378 −0.454224 −0.227112 0.973869i $$-0.572928\pi$$
−0.227112 + 0.973869i $$0.572928\pi$$
$$90$$ 0 0
$$91$$ −160.444 −0.184825
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −2315.47 −2.50065
$$96$$ 0 0
$$97$$ 659.154 0.689969 0.344984 0.938608i $$-0.387884\pi$$
0.344984 + 0.938608i $$0.387884\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 498.510 0.491125 0.245562 0.969381i $$-0.421027\pi$$
0.245562 + 0.969381i $$0.421027\pi$$
$$102$$ 0 0
$$103$$ −196.821 −0.188285 −0.0941425 0.995559i $$-0.530011\pi$$
−0.0941425 + 0.995559i $$0.530011\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 359.378 0.324695 0.162347 0.986734i $$-0.448093\pi$$
0.162347 + 0.986734i $$0.448093\pi$$
$$108$$ 0 0
$$109$$ −1969.73 −1.73088 −0.865441 0.501011i $$-0.832962\pi$$
−0.865441 + 0.501011i $$0.832962\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 693.643 0.577456 0.288728 0.957411i $$-0.406768\pi$$
0.288728 + 0.957411i $$0.406768\pi$$
$$114$$ 0 0
$$115$$ 1826.66 1.48119
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −889.911 −0.685529
$$120$$ 0 0
$$121$$ 1551.49 1.16566
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1646.53 −1.17816
$$126$$ 0 0
$$127$$ 2656.78 1.85631 0.928153 0.372199i $$-0.121396\pi$$
0.928153 + 0.372199i $$0.121396\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 615.734 0.410664 0.205332 0.978692i $$-0.434173\pi$$
0.205332 + 0.978692i $$0.434173\pi$$
$$132$$ 0 0
$$133$$ 2818.22 1.83737
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 613.290 0.382459 0.191230 0.981545i $$-0.438753\pi$$
0.191230 + 0.981545i $$0.438753\pi$$
$$138$$ 0 0
$$139$$ 1899.29 1.15896 0.579480 0.814987i $$-0.303256\pi$$
0.579480 + 0.814987i $$0.303256\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 384.175 0.224660
$$144$$ 0 0
$$145$$ −3791.29 −2.17137
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −976.377 −0.536832 −0.268416 0.963303i $$-0.586500\pi$$
−0.268416 + 0.963303i $$0.586500\pi$$
$$150$$ 0 0
$$151$$ 683.132 0.368162 0.184081 0.982911i $$-0.441069\pi$$
0.184081 + 0.982911i $$0.441069\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −2717.07 −1.40800
$$156$$ 0 0
$$157$$ 511.109 0.259815 0.129907 0.991526i $$-0.458532\pi$$
0.129907 + 0.991526i $$0.458532\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −2223.29 −1.08832
$$162$$ 0 0
$$163$$ −2425.95 −1.16574 −0.582869 0.812566i $$-0.698070\pi$$
−0.582869 + 0.812566i $$0.698070\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −337.332 −0.156309 −0.0781544 0.996941i $$-0.524903\pi$$
−0.0781544 + 0.996941i $$0.524903\pi$$
$$168$$ 0 0
$$169$$ −2145.80 −0.976694
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −2648.29 −1.16385 −0.581924 0.813243i $$-0.697700\pi$$
−0.581924 + 0.813243i $$0.697700\pi$$
$$174$$ 0 0
$$175$$ 4806.82 2.07635
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 2907.29 1.21397 0.606986 0.794713i $$-0.292379\pi$$
0.606986 + 0.794713i $$0.292379\pi$$
$$180$$ 0 0
$$181$$ −3682.80 −1.51238 −0.756188 0.654354i $$-0.772940\pi$$
−0.756188 + 0.654354i $$0.772940\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 2315.47 0.920197
$$186$$ 0 0
$$187$$ 2130.85 0.833277
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1279.56 −0.484740 −0.242370 0.970184i $$-0.577925\pi$$
−0.242370 + 0.970184i $$0.577925\pi$$
$$192$$ 0 0
$$193$$ −4836.84 −1.80396 −0.901978 0.431783i $$-0.857885\pi$$
−0.901978 + 0.431783i $$0.857885\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2869.31 −1.03771 −0.518857 0.854861i $$-0.673642\pi$$
−0.518857 + 0.854861i $$0.673642\pi$$
$$198$$ 0 0
$$199$$ 652.242 0.232343 0.116171 0.993229i $$-0.462938\pi$$
0.116171 + 0.993229i $$0.462938\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 4614.49 1.59544
$$204$$ 0 0
$$205$$ −9329.82 −3.17865
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −6748.08 −2.23337
$$210$$ 0 0
$$211$$ −537.511 −0.175373 −0.0876866 0.996148i $$-0.527947\pi$$
−0.0876866 + 0.996148i $$0.527947\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 7621.06 2.41745
$$216$$ 0 0
$$217$$ 3307.02 1.03454
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 283.997 0.0864421
$$222$$ 0 0
$$223$$ −4041.98 −1.21377 −0.606885 0.794790i $$-0.707581\pi$$
−0.606885 + 0.794790i $$0.707581\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 3070.22 0.897699 0.448850 0.893607i $$-0.351834\pi$$
0.448850 + 0.893607i $$0.351834\pi$$
$$228$$ 0 0
$$229$$ 205.110 0.0591881 0.0295940 0.999562i $$-0.490579\pi$$
0.0295940 + 0.999562i $$0.490579\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −13.3776 −0.00376137 −0.00188068 0.999998i $$-0.500599\pi$$
−0.00188068 + 0.999998i $$0.500599\pi$$
$$234$$ 0 0
$$235$$ −5782.93 −1.60526
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −4327.99 −1.17136 −0.585679 0.810543i $$-0.699172\pi$$
−0.585679 + 0.810543i $$0.699172\pi$$
$$240$$ 0 0
$$241$$ −1508.31 −0.403150 −0.201575 0.979473i $$-0.564606\pi$$
−0.201575 + 0.979473i $$0.564606\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −2943.04 −0.767446
$$246$$ 0 0
$$247$$ −899.378 −0.231684
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −4871.47 −1.22504 −0.612518 0.790456i $$-0.709843\pi$$
−0.612518 + 0.790456i $$0.709843\pi$$
$$252$$ 0 0
$$253$$ 5323.55 1.32288
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 1665.55 0.404258 0.202129 0.979359i $$-0.435214\pi$$
0.202129 + 0.979359i $$0.435214\pi$$
$$258$$ 0 0
$$259$$ −2818.22 −0.676122
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −7167.64 −1.68052 −0.840258 0.542188i $$-0.817596\pi$$
−0.840258 + 0.542188i $$0.817596\pi$$
$$264$$ 0 0
$$265$$ 816.713 0.189322
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −5453.84 −1.23616 −0.618079 0.786116i $$-0.712089\pi$$
−0.618079 + 0.786116i $$0.712089\pi$$
$$270$$ 0 0
$$271$$ −5416.20 −1.21406 −0.607031 0.794678i $$-0.707640\pi$$
−0.607031 + 0.794678i $$0.707640\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −11509.7 −2.52385
$$276$$ 0 0
$$277$$ 2648.75 0.574542 0.287271 0.957849i $$-0.407252\pi$$
0.287271 + 0.957849i $$0.407252\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6664.57 −1.41486 −0.707429 0.706784i $$-0.750145\pi$$
−0.707429 + 0.706784i $$0.750145\pi$$
$$282$$ 0 0
$$283$$ 5630.84 1.18275 0.591376 0.806396i $$-0.298585\pi$$
0.591376 + 0.806396i $$0.298585\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 11355.6 2.33554
$$288$$ 0 0
$$289$$ −3337.80 −0.679381
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 908.374 0.181119 0.0905593 0.995891i $$-0.471135\pi$$
0.0905593 + 0.995891i $$0.471135\pi$$
$$294$$ 0 0
$$295$$ −5968.79 −1.17802
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 709.517 0.137232
$$300$$ 0 0
$$301$$ −9275.82 −1.77624
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 5968.79 1.12056
$$306$$ 0 0
$$307$$ 414.671 0.0770896 0.0385448 0.999257i $$-0.487728\pi$$
0.0385448 + 0.999257i $$0.487728\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −1615.91 −0.294629 −0.147315 0.989090i $$-0.547063\pi$$
−0.147315 + 0.989090i $$0.547063\pi$$
$$312$$ 0 0
$$313$$ −8479.33 −1.53125 −0.765623 0.643289i $$-0.777569\pi$$
−0.765623 + 0.643289i $$0.777569\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −6774.73 −1.20034 −0.600169 0.799873i $$-0.704900\pi$$
−0.600169 + 0.799873i $$0.704900\pi$$
$$318$$ 0 0
$$319$$ −11049.2 −1.93929
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −4988.44 −0.859332
$$324$$ 0 0
$$325$$ −1534.00 −0.261818
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 7038.57 1.17948
$$330$$ 0 0
$$331$$ −9292.36 −1.54306 −0.771532 0.636191i $$-0.780509\pi$$
−0.771532 + 0.636191i $$0.780509\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 8552.80 1.39489
$$336$$ 0 0
$$337$$ 6563.78 1.06098 0.530492 0.847690i $$-0.322007\pi$$
0.530492 + 0.847690i $$0.322007\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −7918.49 −1.25751
$$342$$ 0 0
$$343$$ −4108.75 −0.646798
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −3870.93 −0.598855 −0.299427 0.954119i $$-0.596796\pi$$
−0.299427 + 0.954119i $$0.596796\pi$$
$$348$$ 0 0
$$349$$ −3474.57 −0.532922 −0.266461 0.963846i $$-0.585854\pi$$
−0.266461 + 0.963846i $$0.585854\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 4308.58 0.649639 0.324820 0.945776i $$-0.394696\pi$$
0.324820 + 0.945776i $$0.394696\pi$$
$$354$$ 0 0
$$355$$ 19395.7 2.89977
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 8161.19 1.19981 0.599904 0.800072i $$-0.295205\pi$$
0.599904 + 0.800072i $$0.295205\pi$$
$$360$$ 0 0
$$361$$ 8938.68 1.30320
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −18832.4 −2.70064
$$366$$ 0 0
$$367$$ 4427.66 0.629760 0.314880 0.949132i $$-0.398036\pi$$
0.314880 + 0.949132i $$0.398036\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −994.045 −0.139106
$$372$$ 0 0
$$373$$ −11278.5 −1.56562 −0.782812 0.622258i $$-0.786215\pi$$
−0.782812 + 0.622258i $$0.786215\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −1472.62 −0.201177
$$378$$ 0 0
$$379$$ −709.683 −0.0961846 −0.0480923 0.998843i $$-0.515314\pi$$
−0.0480923 + 0.998843i $$0.515314\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 1233.69 0.164592 0.0822962 0.996608i $$-0.473775\pi$$
0.0822962 + 0.996608i $$0.473775\pi$$
$$384$$ 0 0
$$385$$ 22177.1 2.93571
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −6830.06 −0.890226 −0.445113 0.895474i $$-0.646836\pi$$
−0.445113 + 0.895474i $$0.646836\pi$$
$$390$$ 0 0
$$391$$ 3935.37 0.509003
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 11106.9 1.41481
$$396$$ 0 0
$$397$$ 11289.7 1.42724 0.713618 0.700535i $$-0.247055\pi$$
0.713618 + 0.700535i $$0.247055\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 3055.59 0.380521 0.190261 0.981734i $$-0.439067\pi$$
0.190261 + 0.981734i $$0.439067\pi$$
$$402$$ 0 0
$$403$$ −1055.37 −0.130451
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 6748.08 0.821843
$$408$$ 0 0
$$409$$ 4089.01 0.494349 0.247175 0.968971i $$-0.420498\pi$$
0.247175 + 0.968971i $$0.420498\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 7264.79 0.865562
$$414$$ 0 0
$$415$$ −291.471 −0.0344765
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 15397.0 1.79520 0.897602 0.440806i $$-0.145307\pi$$
0.897602 + 0.440806i $$0.145307\pi$$
$$420$$ 0 0
$$421$$ −1034.45 −0.119753 −0.0598766 0.998206i $$-0.519071\pi$$
−0.0598766 + 0.998206i $$0.519071\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −8508.40 −0.971101
$$426$$ 0 0
$$427$$ −7264.79 −0.823344
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 4943.86 0.552523 0.276261 0.961083i $$-0.410904\pi$$
0.276261 + 0.961083i $$0.410904\pi$$
$$432$$ 0 0
$$433$$ 337.202 0.0374247 0.0187124 0.999825i $$-0.494043\pi$$
0.0187124 + 0.999825i $$0.494043\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −12462.7 −1.36424
$$438$$ 0 0
$$439$$ 4493.93 0.488573 0.244286 0.969703i $$-0.421446\pi$$
0.244286 + 0.969703i $$0.421446\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 4292.26 0.460341 0.230171 0.973150i $$-0.426072\pi$$
0.230171 + 0.973150i $$0.426072\pi$$
$$444$$ 0 0
$$445$$ 7025.82 0.748440
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 4167.96 0.438081 0.219040 0.975716i $$-0.429707\pi$$
0.219040 + 0.975716i $$0.429707\pi$$
$$450$$ 0 0
$$451$$ −27190.4 −2.83890
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 2955.73 0.304543
$$456$$ 0 0
$$457$$ 301.643 0.0308759 0.0154380 0.999881i $$-0.495086\pi$$
0.0154380 + 0.999881i $$0.495086\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 9611.88 0.971085 0.485542 0.874213i $$-0.338622\pi$$
0.485542 + 0.874213i $$0.338622\pi$$
$$462$$ 0 0
$$463$$ −13251.0 −1.33008 −0.665041 0.746807i $$-0.731586\pi$$
−0.665041 + 0.746807i $$0.731586\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 4432.00 0.439161 0.219581 0.975594i $$-0.429531\pi$$
0.219581 + 0.975594i $$0.429531\pi$$
$$468$$ 0 0
$$469$$ −10409.9 −1.02491
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 22210.5 2.15907
$$474$$ 0 0
$$475$$ 26944.9 2.60277
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −9076.49 −0.865794 −0.432897 0.901443i $$-0.642509\pi$$
−0.432897 + 0.901443i $$0.642509\pi$$
$$480$$ 0 0
$$481$$ 899.378 0.0852559
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −12143.1 −1.13688
$$486$$ 0 0
$$487$$ −3343.89 −0.311142 −0.155571 0.987825i $$-0.549722\pi$$
−0.155571 + 0.987825i $$0.549722\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 2423.73 0.222773 0.111386 0.993777i $$-0.464471\pi$$
0.111386 + 0.993777i $$0.464471\pi$$
$$492$$ 0 0
$$493$$ −8167.95 −0.746179
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −23607.1 −2.13063
$$498$$ 0 0
$$499$$ 811.819 0.0728296 0.0364148 0.999337i $$-0.488406\pi$$
0.0364148 + 0.999337i $$0.488406\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −18192.4 −1.61264 −0.806320 0.591479i $$-0.798544\pi$$
−0.806320 + 0.591479i $$0.798544\pi$$
$$504$$ 0 0
$$505$$ −9183.65 −0.809242
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 5645.44 0.491610 0.245805 0.969319i $$-0.420948\pi$$
0.245805 + 0.969319i $$0.420948\pi$$
$$510$$ 0 0
$$511$$ 22921.5 1.98432
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 3625.88 0.310243
$$516$$ 0 0
$$517$$ −16853.5 −1.43369
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −12338.7 −1.03756 −0.518780 0.854908i $$-0.673614\pi$$
−0.518780 + 0.854908i $$0.673614\pi$$
$$522$$ 0 0
$$523$$ −10609.8 −0.887062 −0.443531 0.896259i $$-0.646274\pi$$
−0.443531 + 0.896259i $$0.646274\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −5853.65 −0.483850
$$528$$ 0 0
$$529$$ −2335.17 −0.191926
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −3623.91 −0.294501
$$534$$ 0 0
$$535$$ −6620.53 −0.535010
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −8577.07 −0.685419
$$540$$ 0 0
$$541$$ 4035.42 0.320696 0.160348 0.987061i $$-0.448738\pi$$
0.160348 + 0.987061i $$0.448738\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 36286.8 2.85203
$$546$$ 0 0
$$547$$ 7407.45 0.579012 0.289506 0.957176i $$-0.406509\pi$$
0.289506 + 0.957176i $$0.406509\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 25866.7 1.99993
$$552$$ 0 0
$$553$$ −13518.6 −1.03954
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 9500.77 0.722730 0.361365 0.932424i $$-0.382311\pi$$
0.361365 + 0.932424i $$0.382311\pi$$
$$558$$ 0 0
$$559$$ 2960.19 0.223976
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −2700.26 −0.202136 −0.101068 0.994880i $$-0.532226\pi$$
−0.101068 + 0.994880i $$0.532226\pi$$
$$564$$ 0 0
$$565$$ −12778.4 −0.951492
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −15904.9 −1.17183 −0.585913 0.810374i $$-0.699264\pi$$
−0.585913 + 0.810374i $$0.699264\pi$$
$$570$$ 0 0
$$571$$ 18234.0 1.33638 0.668188 0.743992i $$-0.267070\pi$$
0.668188 + 0.743992i $$0.267070\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −21256.7 −1.54168
$$576$$ 0 0
$$577$$ 4869.57 0.351339 0.175670 0.984449i $$-0.443791\pi$$
0.175670 + 0.984449i $$0.443791\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 354.758 0.0253319
$$582$$ 0 0
$$583$$ 2380.19 0.169086
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 1616.99 0.113697 0.0568485 0.998383i $$-0.481895\pi$$
0.0568485 + 0.998383i $$0.481895\pi$$
$$588$$ 0 0
$$589$$ 18537.7 1.29683
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −8117.01 −0.562101 −0.281050 0.959693i $$-0.590683\pi$$
−0.281050 + 0.959693i $$0.590683\pi$$
$$594$$ 0 0
$$595$$ 16394.1 1.12957
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −9536.40 −0.650495 −0.325248 0.945629i $$-0.605448\pi$$
−0.325248 + 0.945629i $$0.605448\pi$$
$$600$$ 0 0
$$601$$ −16247.8 −1.10276 −0.551381 0.834253i $$-0.685899\pi$$
−0.551381 + 0.834253i $$0.685899\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −28581.9 −1.92069
$$606$$ 0 0
$$607$$ 27725.7 1.85396 0.926980 0.375111i $$-0.122395\pi$$
0.926980 + 0.375111i $$0.122395\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −2246.22 −0.148727
$$612$$ 0 0
$$613$$ 927.190 0.0610911 0.0305456 0.999533i $$-0.490276\pi$$
0.0305456 + 0.999533i $$0.490276\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18727.8 1.22196 0.610982 0.791644i $$-0.290775\pi$$
0.610982 + 0.791644i $$0.290775\pi$$
$$618$$ 0 0
$$619$$ 3210.22 0.208449 0.104224 0.994554i $$-0.466764\pi$$
0.104224 + 0.994554i $$0.466764\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −8551.33 −0.549922
$$624$$ 0 0
$$625$$ 3535.57 0.226276
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 4988.44 0.316220
$$630$$ 0 0
$$631$$ −11911.2 −0.751468 −0.375734 0.926728i $$-0.622609\pi$$
−0.375734 + 0.926728i $$0.622609\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −48943.7 −3.05869
$$636$$ 0 0
$$637$$ −1143.14 −0.0711036
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −17232.6 −1.06185 −0.530924 0.847419i $$-0.678155\pi$$
−0.530924 + 0.847419i $$0.678155\pi$$
$$642$$ 0 0
$$643$$ −12754.6 −0.782262 −0.391131 0.920335i $$-0.627916\pi$$
−0.391131 + 0.920335i $$0.627916\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −9441.73 −0.573714 −0.286857 0.957973i $$-0.592610\pi$$
−0.286857 + 0.957973i $$0.592610\pi$$
$$648$$ 0 0
$$649$$ −17395.2 −1.05211
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 5198.99 0.311565 0.155783 0.987791i $$-0.450210\pi$$
0.155783 + 0.987791i $$0.450210\pi$$
$$654$$ 0 0
$$655$$ −11343.2 −0.676664
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −6508.01 −0.384698 −0.192349 0.981327i $$-0.561611\pi$$
−0.192349 + 0.981327i $$0.561611\pi$$
$$660$$ 0 0
$$661$$ 25280.2 1.48757 0.743785 0.668419i $$-0.233028\pi$$
0.743785 + 0.668419i $$0.233028\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −51917.8 −3.02750
$$666$$ 0 0
$$667$$ −20406.2 −1.18460
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 17395.2 1.00079
$$672$$ 0 0
$$673$$ −5525.89 −0.316504 −0.158252 0.987399i $$-0.550586\pi$$
−0.158252 + 0.987399i $$0.550586\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 6293.21 0.357264 0.178632 0.983916i $$-0.442833\pi$$
0.178632 + 0.983916i $$0.442833\pi$$
$$678$$ 0 0
$$679$$ 14779.7 0.835335
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 5675.91 0.317984 0.158992 0.987280i $$-0.449176\pi$$
0.158992 + 0.987280i $$0.449176\pi$$
$$684$$ 0 0
$$685$$ −11298.2 −0.630190
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 317.229 0.0175406
$$690$$ 0 0
$$691$$ −3617.79 −0.199171 −0.0995854 0.995029i $$-0.531752\pi$$
−0.0995854 + 0.995029i $$0.531752\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −34989.1 −1.90966
$$696$$ 0 0
$$697$$ −20100.2 −1.09232
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 7938.43 0.427718 0.213859 0.976865i $$-0.431397\pi$$
0.213859 + 0.976865i $$0.431397\pi$$
$$702$$ 0 0
$$703$$ −15797.7 −0.847540
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 11177.7 0.594597
$$708$$ 0 0
$$709$$ −25691.6 −1.36088 −0.680442 0.732802i $$-0.738212\pi$$
−0.680442 + 0.732802i $$0.738212\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −14624.3 −0.768142
$$714$$ 0 0
$$715$$ −7077.35 −0.370179
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −36803.3 −1.90895 −0.954473 0.298298i $$-0.903581\pi$$
−0.954473 + 0.298298i $$0.903581\pi$$
$$720$$ 0 0
$$721$$ −4413.16 −0.227954
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 44118.9 2.26005
$$726$$ 0 0
$$727$$ 28333.2 1.44542 0.722709 0.691152i $$-0.242897\pi$$
0.722709 + 0.691152i $$0.242897\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 16418.8 0.830742
$$732$$ 0 0
$$733$$ −10767.9 −0.542592 −0.271296 0.962496i $$-0.587452\pi$$
−0.271296 + 0.962496i $$0.587452\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 24925.9 1.24580
$$738$$ 0 0
$$739$$ −17301.4 −0.861221 −0.430610 0.902538i $$-0.641702\pi$$
−0.430610 + 0.902538i $$0.641702\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −24110.8 −1.19050 −0.595248 0.803542i $$-0.702946\pi$$
−0.595248 + 0.803542i $$0.702946\pi$$
$$744$$ 0 0
$$745$$ 17987.0 0.884555
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 8058.04 0.393103
$$750$$ 0 0
$$751$$ 30052.8 1.46024 0.730121 0.683318i $$-0.239464\pi$$
0.730121 + 0.683318i $$0.239464\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −12584.8 −0.606633
$$756$$ 0 0
$$757$$ 25599.4 1.22910 0.614549 0.788879i $$-0.289338\pi$$
0.614549 + 0.788879i $$0.289338\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 28904.5 1.37685 0.688427 0.725306i $$-0.258302\pi$$
0.688427 + 0.725306i $$0.258302\pi$$
$$762$$ 0 0
$$763$$ −44165.7 −2.09555
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −2318.41 −0.109143
$$768$$ 0 0
$$769$$ −8756.13 −0.410603 −0.205302 0.978699i $$-0.565818\pi$$
−0.205302 + 0.978699i $$0.565818\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −28289.8 −1.31632 −0.658159 0.752879i $$-0.728665\pi$$
−0.658159 + 0.752879i $$0.728665\pi$$
$$774$$ 0 0
$$775$$ 31618.3 1.46550
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 63654.4 2.92767
$$780$$ 0 0
$$781$$ 56526.0 2.58983
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −9415.75 −0.428105
$$786$$ 0 0
$$787$$ 4859.74 0.220116 0.110058 0.993925i $$-0.464896\pi$$
0.110058 + 0.993925i $$0.464896\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 15553.0 0.699117
$$792$$ 0 0
$$793$$ 2318.41 0.103820
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −17361.7 −0.771623 −0.385811 0.922578i $$-0.626078\pi$$
−0.385811 + 0.922578i $$0.626078\pi$$
$$798$$ 0 0
$$799$$ −12458.8 −0.551638
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −54884.2 −2.41198
$$804$$ 0 0
$$805$$ 40957.8 1.79326
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 24475.3 1.06367 0.531833 0.846849i $$-0.321503\pi$$
0.531833 + 0.846849i $$0.321503\pi$$
$$810$$ 0 0
$$811$$ 19875.4 0.860566 0.430283 0.902694i $$-0.358414\pi$$
0.430283 + 0.902694i $$0.358414\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 44691.4 1.92083
$$816$$ 0 0
$$817$$ −51996.1 −2.22658
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −21682.1 −0.921693 −0.460846 0.887480i $$-0.652454\pi$$
−0.460846 + 0.887480i $$0.652454\pi$$
$$822$$ 0 0
$$823$$ 6698.17 0.283698 0.141849 0.989888i $$-0.454695\pi$$
0.141849 + 0.989888i $$0.454695\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 3390.85 0.142577 0.0712886 0.997456i $$-0.477289\pi$$
0.0712886 + 0.997456i $$0.477289\pi$$
$$828$$ 0 0
$$829$$ −40093.4 −1.67974 −0.839869 0.542789i $$-0.817368\pi$$
−0.839869 + 0.542789i $$0.817368\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −6340.50 −0.263728
$$834$$ 0 0
$$835$$ 6214.40 0.257555
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 6172.12 0.253975 0.126988 0.991904i $$-0.459469\pi$$
0.126988 + 0.991904i $$0.459469\pi$$
$$840$$ 0 0
$$841$$ 17964.6 0.736585
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 39530.3 1.60933
$$846$$ 0 0
$$847$$ 34787.8 1.41124
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 12462.7 0.502018
$$852$$ 0 0
$$853$$ −276.632 −0.0111040 −0.00555198 0.999985i $$-0.501767\pi$$
−0.00555198 + 0.999985i $$0.501767\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −3704.41 −0.147655 −0.0738274 0.997271i $$-0.523521\pi$$
−0.0738274 + 0.997271i $$0.523521\pi$$
$$858$$ 0 0
$$859$$ 26915.5 1.06909 0.534544 0.845141i $$-0.320484\pi$$
0.534544 + 0.845141i $$0.320484\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 23623.9 0.931828 0.465914 0.884830i $$-0.345726\pi$$
0.465914 + 0.884830i $$0.345726\pi$$
$$864$$ 0 0
$$865$$ 48787.3 1.91771
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 32369.5 1.26359
$$870$$ 0 0
$$871$$ 3322.10 0.129236
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −36918.9 −1.42638
$$876$$ 0 0
$$877$$ 14094.0 0.542667 0.271333 0.962485i $$-0.412535\pi$$
0.271333 + 0.962485i $$0.412535\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −18967.3 −0.725341 −0.362671 0.931917i $$-0.618135\pi$$
−0.362671 + 0.931917i $$0.618135\pi$$
$$882$$ 0 0
$$883$$ 32886.0 1.25334 0.626672 0.779283i $$-0.284417\pi$$
0.626672 + 0.779283i $$0.284417\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 27945.6 1.05786 0.528929 0.848666i $$-0.322594\pi$$
0.528929 + 0.848666i $$0.322594\pi$$
$$888$$ 0 0
$$889$$ 59570.8 2.24740
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 39455.1 1.47852
$$894$$ 0 0
$$895$$ −53558.6 −2.00030
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 30353.1 1.12607
$$900$$ 0 0
$$901$$ 1759.53 0.0650592
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 67845.2 2.49199
$$906$$ 0 0
$$907$$ −5544.89 −0.202993 −0.101497 0.994836i $$-0.532363\pi$$
−0.101497 + 0.994836i $$0.532363\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −19638.9 −0.714233 −0.357116 0.934060i $$-0.616240\pi$$
−0.357116 + 0.934060i $$0.616240\pi$$
$$912$$ 0 0
$$913$$ −849.451 −0.0307916
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 13806.1 0.497184
$$918$$ 0 0
$$919$$ 22128.7 0.794297 0.397149 0.917754i $$-0.370000\pi$$
0.397149 + 0.917754i $$0.370000\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 7533.72 0.268663
$$924$$ 0 0
$$925$$ −26944.9 −0.957775
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −17599.3 −0.621544 −0.310772 0.950484i $$-0.600588\pi$$
−0.310772 + 0.950484i $$0.600588\pi$$
$$930$$ 0 0
$$931$$ 20079.5 0.706850
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −39254.9 −1.37302
$$936$$ 0 0
$$937$$ 441.299 0.0153859 0.00769297 0.999970i $$-0.497551\pi$$
0.00769297 + 0.999970i $$0.497551\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −5408.01 −0.187350 −0.0936749 0.995603i $$-0.529861\pi$$
−0.0936749 + 0.995603i $$0.529861\pi$$
$$942$$ 0 0
$$943$$ −50216.8 −1.73413
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 34237.1 1.17482 0.587410 0.809289i $$-0.300148\pi$$
0.587410 + 0.809289i $$0.300148\pi$$
$$948$$ 0 0
$$949$$ −7314.92 −0.250213
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −21410.7 −0.727766 −0.363883 0.931445i $$-0.618549\pi$$
−0.363883 + 0.931445i $$0.618549\pi$$
$$954$$ 0 0
$$955$$ 23572.2 0.798722
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 13751.3 0.463038
$$960$$ 0 0
$$961$$ −8038.09 −0.269816
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 89105.3 2.97243
$$966$$ 0 0
$$967$$ −53874.6 −1.79161 −0.895806 0.444445i $$-0.853401\pi$$
−0.895806 + 0.444445i $$0.853401\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −42901.5 −1.41789 −0.708947 0.705262i $$-0.750829\pi$$
−0.708947 + 0.705262i $$0.750829\pi$$
$$972$$ 0 0
$$973$$ 42586.2 1.40314
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −58636.5 −1.92011 −0.960055 0.279812i $$-0.909728\pi$$
−0.960055 + 0.279812i $$0.909728\pi$$
$$978$$ 0 0
$$979$$ 20475.7 0.668444
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 25296.7 0.820793 0.410396 0.911907i $$-0.365390\pi$$
0.410396 + 0.911907i $$0.365390\pi$$
$$984$$ 0 0
$$985$$ 52859.0 1.70988
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 41019.6 1.31885
$$990$$ 0 0
$$991$$ 10605.2 0.339944 0.169972 0.985449i $$-0.445632\pi$$
0.169972 + 0.985449i $$0.445632\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −12015.7 −0.382839
$$996$$ 0 0
$$997$$ −5770.19 −0.183294 −0.0916469 0.995792i $$-0.529213\pi$$
−0.0916469 + 0.995792i $$0.529213\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2304.4.a.t.1.1 2
3.2 odd 2 768.4.a.j.1.2 2
4.3 odd 2 2304.4.a.s.1.1 2
8.3 odd 2 2304.4.a.bp.1.2 2
8.5 even 2 2304.4.a.bq.1.2 2
12.11 even 2 768.4.a.p.1.2 2
16.3 odd 4 1152.4.d.o.577.4 4
16.5 even 4 1152.4.d.i.577.1 4
16.11 odd 4 1152.4.d.o.577.1 4
16.13 even 4 1152.4.d.i.577.4 4
24.5 odd 2 768.4.a.k.1.1 2
24.11 even 2 768.4.a.e.1.1 2
48.5 odd 4 384.4.d.c.193.4 yes 4
48.11 even 4 384.4.d.e.193.2 yes 4
48.29 odd 4 384.4.d.c.193.1 4
48.35 even 4 384.4.d.e.193.3 yes 4

By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.c.193.1 4 48.29 odd 4
384.4.d.c.193.4 yes 4 48.5 odd 4
384.4.d.e.193.2 yes 4 48.11 even 4
384.4.d.e.193.3 yes 4 48.35 even 4
768.4.a.e.1.1 2 24.11 even 2
768.4.a.j.1.2 2 3.2 odd 2
768.4.a.k.1.1 2 24.5 odd 2
768.4.a.p.1.2 2 12.11 even 2
1152.4.d.i.577.1 4 16.5 even 4
1152.4.d.i.577.4 4 16.13 even 4
1152.4.d.o.577.1 4 16.11 odd 4
1152.4.d.o.577.4 4 16.3 odd 4
2304.4.a.s.1.1 2 4.3 odd 2
2304.4.a.t.1.1 2 1.1 even 1 trivial
2304.4.a.bp.1.2 2 8.3 odd 2
2304.4.a.bq.1.2 2 8.5 even 2